Problem: You have found the following ages (in years) of all 4 meerkats at your local zoo: $ 8,\enspace 14,\enspace 3,\enspace 3$ What is the average age of the meerkats at your zoo? What is the standard deviation? You may round your answers to the nearest tenth.
Solution: Because we have data for all 4 meerkats at the zoo, we are able to calculate the population mean $({\mu})$ and population standard deviation $({\sigma})$ To find the population mean , add up the values of all $4$ ages and divide by $4$ $ {\mu} = \dfrac{\sum\limits_{i=1}^{{N}} x_i}{{N}} = \dfrac{\sum\limits_{i=1}^{{4}} x_i}{{4}} $ $ {\mu} = \dfrac{8 + 14 + 3 + 3}{{4}} = {7\text{ years old}} $ Find the squared deviations from the mean for each meerkat. Age $x_i$ Distance from the mean $(x_i - {\mu})$ $(x_i - {\mu})^2$ $8$ years $1$ year $1$ year $^2$ $14$ years $7$ years $49$ years $^2$ $3$ years $-4$ years $16$ years $^2$ $3$ years $-4$ years $16$ years $^2$ Because we used the population mean $({\mu})$ to compute the squared deviations from the mean , we can find the variance $({\sigma^2})$ , without introducing any bias, by simply averaging the squared deviations from the mean $ {\sigma^2} = \dfrac{\sum\limits_{i=1}^{{N}} (x_i - {\mu})^2}{{N}} $ $ {\sigma^2} = \dfrac{{1} + {49} + {16} + {16}} {{4}} $ $ {\sigma^2} = \dfrac{{82}}{{4}} = {20.5\text{ years}^2} $ As you might guess from the notation, the population standard deviation $({\sigma})$ is found by taking the square root of the population variance $({\sigma^2})$ ${\sigma} = \sqrt{{\sigma^2}}$ $ {\sigma} = \sqrt{{20.5\text{ years}^2}} = {4.5\text{ years}} $ The average meerkat at the zoo is 7 years old. There is a standard deviation of 4.5 years.